It is proved that every connected graph G on n vertices with $\chi (G)\ge 4$ has at most $k{(k-1)}^{n-3}(k-2)(k-3)$ k-colourings for every $k\ge 4$. Equality holds for some (and then for every) k if and only if the graph is formed from $K_4$ by repeatedly adding leaves. This confirms (a strengthening of) the 4-chromatic case of a long-standing conjecture of Tomescu [29]. Proof methods may be of independent interest. In particular, one of our auxiliary results about list-chromatic polynomials solves a generalized version of a recent conjecture of Brown, Erey, and Li.

中文翻译：

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最大着色数：四色图

证明在n个顶点上的每个连通图G具有$\chi （G）\ge 4$ 最多 $ķ{（ķ-\mathrm{1个}）}^{ñ-3}（ķ-2）（ķ-3）$ 每个K色$ķ\ge 4$。等号成立对于一些（然后为每一个）ķ当且仅当该图从形成${ķ}_4$通过反复添加叶子。这证实了（增强）Tomescu [29]长期存在的猜想的四色情况。证明方法可能具有独立利益。特别是，我们有关列表色多项式的辅助结果之一解决了Brown，Erey和Li最近猜想的广义形式。